Introduction To Recurring Fractions


I. Introduction
What happens when you try to convert the fraction $\dfrac{1}{3}$ to decimal? Let's see.

$\begin{array}{rl}0.333 \\[-5pt]3 \enclose{longdiv}{1.000} \\ \underline{\phantom{0}9\phantom{00}} \\ 10\phantom{0} \\ \underline{\phantom{0}9\phantom{0}} \\ \phantom{0}10 \\ \underline{\phantom{00}9} \\ \phantom{00}1\end{array}$

As you can see, the remainder of the division operation repeats each time you perform the division. You can see that the same pattern will continue for ever, no matter how many times you perform the division. So you will never be able to reach a state where your remainder is zero. This is called a recurring fraction.

As you will see going ahead, that during a long division, after you have reached the non-significant zeros after decimal point, if any of the remainder is repeated then, it is a recurring fraction. Let us take a look at a division by $7$. We will convert $\dfrac{3}{7}$ to decimal.

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$\begin{array}{rl} 0.4285714 \\[-5pt]7 \enclose{longdiv}{3.0000000} \\ \underline{28\phantom{000000}} \\ 20\phantom{00000} & \Leftarrow \hbox{Here the remainder is 2} \\ \underline{14\phantom{00000}} \\ \phantom{0}60\phantom{0000} \\ \underline{56\phantom{0000}} \\ \phantom{00}40\phantom{000}\\ \underline{35\phantom{000}} \\ 50\phantom{00}\\ \underline{49\phantom{00}} \\ 10\phantom{0} \\ \underline{7\phantom{0}} \\ 30 \\ \underline{28} \\ 2 & \Leftarrow \hbox{Remainder 2 repeats here} \end{array}$

We can stop our calculation right when we see that the remainder has repeated. Let's see the result we got

$\dfrac{3}{7} = 0.4285714 \ldots$

Now, we see that the quotient for which the remainder two repeated, is $4$. So we know that the pattern after the first four, till the second $4$ is going to repeat. When writing a recurring decimal, we normally draw a vinculum above the repeating part of the fraction.

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In our case of $\dfrac{3}{7}$  we can write:

$\dfrac{3}{7} = 0.\overline{428571}$
to indicate that the whole pattern $423571$ keeps repeating after the decimal point.
In our first example of $\dfrac{1}{3}$ we can simply write:

$\dfrac{1}{3} = 0.\overline{3}$
to indicate the 3 repeats indefinitely after the decimal point.